Curved quantum waveguides in uniform magnetic fields

Abstract

A theoretical study of a planar electronic waveguide with a uniformly curved section in the perpendicular homogeneous magnetic field $\mathbf{B}$ is presented within the envelope function approximation. Utilizing analytical solutions in each part of the waveguide, exact expressions are derived for the scattering and reflection matrices and for the transcendental equation defining bound-state energies. It is shown that in the magnetic field a propagation threshold in the continuously curved channel is always smaller than its counterpart for the straight arm which means that bound states in the uniform magnetic field always exist. Their energies do not depend on the direction of the field, and at high magnetic intensities they approach the lowest Landau level. For the transport in the fundamental mode an interaction of a quasibound level split off from the higher-lying threshold as a result of the bend, with its degenerate continuum counterpart, causes a dip in the transmission. In the magnetic field, contrary to the field-free case, conductance in the minimum ${G}_{\mathit{min}}$, generally, ceases to be zero. It is shown that growing magnetic fields cause ${G}_{\mathit{min}}$ to saturate to $2{e}^{2}∕h$ which means that a quasibound level formed as a result of the bend is completely dissolved by the increasing $\mathbf{B}$; however, this transformation is very different for the different bend angles and radii. In particular, quasibound states of the fundamental propagation mode survive stronger fields for the smaller bend angles which is explained by the larger total magnetic flux through the curved section where these levels are formed. Since a magnetic length ${l}_{B}={(\ensuremath{\hbar}∕eB)}^{1∕2}$ is inversely proportional to the square root of $B$, states for the waveguide with a smaller radius also survive stronger fields, and their asymptotic approach to the dissolution possesses nonmonotonic ${G}_{\mathit{min}}$ dependence on the magnetic field with minimum conductance again reaching zero for some special values of $B$. Vortex structure of the currents flowing in the waveguide near the resonance is strongly affected by the field. In particular, small magnetic intensities change zero-field vortices in the straight arms into the magnetic antivortices which correspond to the interacting with each other surface currents flowing along opposite walls of the channel. Increasing the magnetic field suppresses the formation of the vortices pushing currents to the outer (inner) walls in the straight (bent) section. For fields larger than the saturation magnetic intensity, the only consequence of the bend is a strong surface current near the convex wall of the bend, and the electronic flow along the junctions between straight and curved parts.